Total Distance vs Displacement

Calculus 1 · Axiom Academy

Total Distance vs Displacement Drive forward, then reverse — and watch two perfectly reasonable answers to how far? pull apart. You drive a car 120 m forward down a straight road, then back up 30 m . Your odometer reads 150 m of ground covered — but you re only 90 m from where you started. One trip, two different how far? — that gap is the whole idea. The car always goes 120 m forward first. Set how far it then reverses , hit Go, and watch the two readouts. Displacement is where it ends up; total distance is every metre the tyres turned. Read it off the velocity graph In calculus the two answers come from the same velocity curve, split by one symbol. Here v(t) = 20 - 5t — the car slows, stops at t = 4 , then runs backward. Drag the clock: displacement is the signed area _0^ T v\,dt (red below the axis subtracts); total distance is the unsigned area _0^ T v \,dt . Two devices ride along on every trip. The odometer only ever adds up motion — that s v \,dt . The GPS distance from home tracks your net displacement — v\,dt . Drive 150 m out, then drag yourself back toward home and watch them disagree. One velocity, one integral, one symbol of difference: displacement = v\,dt lets backward motion cancel forward; total distance = v \,dt counts every metre. It s why your odometer outruns the GPS , why a fuel tank cares about distance not displacement, and why a lap of the track leaves you 0 m from the start having run a full mile.

This is the written version of the interactive lesson above. See the full Calculus 1 course.